Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,8)(7,10,11,12)(13,14)(15,21)(16,20)(17,18)(19,22)(23,27)(24,25)(26,28)(29,30,34,32)(31,36,39,35)(33,40,42,37)(38,43,41,44);(7,11)(10,12)(14,18)(15,22)(16,24)(26,27)(29,31,34,39)(30,35,32,36)(33,41,42,38)(37,44,40,43);(5,7,9,11)(6,10,8,12)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,28)(25,27)(29,31)(30,36)(32,35)(33,38)(34,39)(37,44)(40,43)(41,42);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,14)(15,21)(16,25)(17,18)(19,22)(20,24)(23,26)(27,28)(29,32,34,30)(31,36,39,35)(33,37,42,40)(38,43,41,44);(5,8,9,6)(7,12,11,10)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)(29,33,34,42)(30,37,32,40)(31,38,39,41)(35,43,36,44)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fh,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2992719);